Valuation ring and localizations

Question

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent.
(1) $D_P$ is a valuation ring for each proper prime $P$ in $D$.
(2) $D_M$ is a valuation ring for each maximal ideal $M$ in $D$.

How do I prove this? I tried working with the definitions, but it didn't turn out to be helpful. Help.

Answer 1

Let's take as definition of a valuation ring $V$ with field of fractions $K$ that for each $u \in K$, either $u \in V$ or $u^{-1} \in V$.

Fact: Let $V$ be a valuation ring with field of fractions $K$. Let $R$ be a ring $V \subseteq R \subseteq K$. Then $R$ is a valuation ring.

Proof of fact is immediate.

(1) -> (2) is obvious since maximal ideals are prime.

(2) -> (1) Let $\mathfrak{p}$ be a prime ideal. It is contained in some maximal ideal $\mathfrak{m}$. We have naturally that $D \subseteq D_\mathfrak{m} \subseteq D_\mathfrak{p} \subseteq K$. So $D_\mathfrak{p}$ is a valuation ring by the above fact and the assumption that $D_\mathfrak{m}$ is a valuation ring.